Transition form factors and helicity amplitudes for electroexcitation of negative- and positive parity nucleon resonances in a light-front quark model

A workable basis of quark configurations $s^3$, $s^2p$ and $sp^2$ at light front has been constructed to describe the high-$Q^2$ behavior of transition form factors and helicity amplitudes in the electroproduction of the lightest nucleon resonances, $N_{1/2^-}(1535)$ and $N_{1/2^+}(1440)$. High-quality data of the CLAS Collaboration are described in the framework of a model which takes into account mixing of the quark configurations and the hadron-molecular states. The model allows for a rough estimate of the quark core weight in the wave function of the resonance in a comparison with high momentum transfer data on resonance electroproduction.


II. FORMALISM
We have taken the formalism developed in Refs. [49,50] as a starting point for our study of light front quark configuration. In this section we represent only basic formulas and definitions of the formalism [49,50] that will be very useful for the short presentation of our results in the following sections. We use notations which are very close to those used in Refs. [49,50].

A. Definitions and notations
Quark state vectors |(m i , s i ); p i , µ i are defined as the basis states of an unitary irreducible representation of the Poincaré group characterized by two invariants, m 2 i and s i (s i +1), which are the proper values of operators M 2 = P µ P µ (square of the mass) and − 1 M 2 W µ W µ (square of spin). The 4-vector W µ is the Pauli-Lyubansky vector and P µ and J µν are generators of the Poincaré group. In the case of a three-quark system one can use the equations P = p 1 + p 2 + p 3 , P 0 = √ P 2 + M 2 , where p µ i is a quark momentum on its mass shell p 0 i = ω i (p i ) := p 2 i + m 2 i . Starting from the direct products of these "plane-wave" quark states 3 i=1 |(m i , s i ); p i , µ i , we can construct the twoand three-quark basis vectors |[m 12 , j 12 (l 12 , s 12 )]; P 12 , µ 12 , |[M 0 , j((j 12 (l 12 , s 12 ), s 3 ), l 3 )]; P , µ j . ( These states have definite values for the orbital angular momentum (l 12 ), the spin (s 12 ) and the sum of them j 12 = l 12 + s 12 for two-quark clusters (P 12 = p 1 + p 2 ) and a definite value for the total angular momentum j = j 12 + s 3 + l 3 of the three-quark system. Here m 12 and M 0 are masses of two-and three-quark free states, respectively. Two-particle basis vectors of the irreducible representation j 12 (l 12 , s 12 ) of the rotation group can be constructed [45] in the rest frame, where P 12 = • P 12 := 0, using standard methods of nonrelativistic quantum mechanics (the Clebsch-Gordan coefficients and spherical functions Y lµ l (p 1 )). Setting up the basis of the three-particle irreducible representation j((j 12 (l 12 , s 12 ), s 3 ), l 3 ) with the same method requires to pass into the three-particle rest frame, where P = • P := 0, but P 12 = 0. This requires a relativistic boost on the two-quark cluster to transform its wave function into the moving (with the 4-velocity P µ 12 /m 12 ) frame. The construction of the irreducible representations of the Poincaré group is performed in the centre of mass (CM) frame where P µ = {M, 0}. A special role of the rest frame in construction of the basis of irreducible representations of the Poincaré group stems from the fact that only in this frame the 4-vector of spin given in Eq. (1) reduces to 3-vector 1 M W µ = {0, J 23 , J 31 , J 12 } which coincides with the 3-vector of rotation generators J ij . Thus one can use a standard technique of the rotation group to construct the basis vectors. The inner relative momenta of a baryon can be specified by the quark momenta k i , i = 1,2.3 in the baryon CM frame, k 1 + k 2 + k 3 = • P := 0 (we use letters k or K for the relative momenta as done in the literature [44,49,50]). The inner relative momenta of the two-quark cluster are specified by the quark momenta k ′ 1 and k ′ 2 in its rest frame ( where Λ( p m ) ≡ Λ µ ν is the matrix of the Lorentz transformation that describes the transition from the two-quark rest frame to the moving frame (the value is a 4-velocity of the two-quark cluster in the baryon CM frame). The 3-momentum k defined by Eq. (3) is one of two independent relative momenta in the three-quark system. A second independent relative momentum may be identified with the momentum K := k 3 = −(k 1 +k 2 ). Masses m 12 and M 0 of the two-and three-quark clusters in the baryon, are functions of two independent relative momenta k and K. The state vector of the baryon in its rest frame (P = • P ) may be symbolically (we omit isospin and other details) represented in terms of a superposition of free basis vectors (2) |(M, j); Φ Mj is a wave function that should depend on an invariant combination of two relative momenta, k and K. The free mass M 0 defined in Eq.(4) may be used as such an invariant combination. For example, the wave function Φ Mj (M 0 ) could be a solution of the three-particle relativistic equation in the framework of the Bakamjian-Thomas [55] approach or it could be a phenomenological wave function. The important property of the integrand in r.h.s. of Eq. (5) is that the three-quark basis state, denoted by proper values of orbital/total angular momenta, can be represented as a superposition of free three-quark plane-wave states 3 i=1 |(m i , s i ); p i , µ i (see later for details) which itself realizes the irreducible representation of the Poincaré group. Therefore the transformation of the state vector (5) into a moving reference frame • P → P = 0 can be readily done by the unitary representation U [Λ g ( P M0 )] of the one-particle boost Λ g ( P M0 ) in plane-wave basis |(m i , s i ); p i , µ i Here P M0 is the spatial part of the 4-velocity u µ = 1 M0 {P 0 ,P } and the index g above specifies the little group used for the transition • P → P (see definitions of the canonical and front boosts in Appendix A).

B. Two-particle states and Melosh transformations
The basis state vectors of the irreducible representation j(ls) of the rotation group can be constructed in the rest frame of the two-particle cluster with the standard non-relativistic technique of adding angular momenta (Clebsch-Gordan coefficients and spherical functions) [45,49,50]: where {µ} := µ 1 , µ 2 , µ s , µ l and m 12 is a mass defined in Eqs. (4) and (A13). The state vector of the physical two-particle system (e.g., a bound state) can be expanded in the basis (6) and represented in form of where m d is the mass of the bound state and Φ j(ls) is the wave function. The canonical basis vectors | ± k, µ i c in the r.h.s. of Eq. (6) can be transformed into the moving reference frame • P 12 → P 12 by making use of the transformation formula of Eq. (A6). But such a transformation is complicated by the Wigner rotation which depends on both the initial and finite momenta of the i-th quark. Thus, it would be the more convenient to pass to the front form of the state vector (7) where R f c (k i ) is the space rotation which connects the front spin of the quark and its canonical spin. In the case of s i = 1 2 the respective D matrix is equal to the matrix element whereμ i and µ i are the z-components of the front and canonical spins, respectively. The final expression for the basis vector (6) in the moving reference frame is of the form wherep 1 = Λ f (P 12 m12 )k,p 2 = Λ f (P 12 m12 )(−k). The components of the 3-vector of the relativistic relative momentum k j = {k ⊥ , k z } are also expressed in terms of invariants, k z : ,k = k |k| . Since the wave function Φ j(ls) and its argument m 12 are relativistic invariants, the expression for the state vector (7) in a moving reference frame, P + 12 m d |[m d , j(l, s)];P 12 , µ j f , can be obtained by the substitution |[m 12 , j(l, s)]; • P 12 , µ j → P + 12 m12 |[m 12 , j(l, s)];P 12 , µ j f in the r.h.s. of Eq. (7).

III. THREE-PARTICLE BASIS STATES
Here we consider three-quark configurations at the light front for cases when the total orbital angular momentum is not larger than l = 1. Then there are three simple variants: {l = 0(l 12 = l 3 = l)}, {l = 0(l 12 = l 3 = 1)} and {l = 1[(l 12 = 0, l 3 = l), (l 12 = l, l 3 = 0)]}. The more complicated variant {l = 1(l 12 = l 3 = l)} is omitted as here we only consider the lowest excited state for each given parity P = ±. This is the minimal basis to evaluate the transition form factors for the low-lying resonances N 1/2 + (1440) and     We construct the three-quark basis states following step by step the method developed in Sect. II for the two-quark state vectors. In the case of low angular momenta l = 0, 1 the three-quark basis vectors are of the same form as the two-quark states given in Eqs. (6) and (10)). Starting from these expressions one can at once write the three-quark basis state having the quantum numbers of the TISM configuration s 2 p[21] X (l = 1)y (1) X (i.e. l 12 = 0, l 3 = 1): Here β is an arbitrary scale (the nucleon inverse radius as usual). The three-quark LF state vector analogous to the TISM configuration s 2 p[21] X (l = 1)y (1) X is defined by an expression which is a replica of Eq. (7): where the wave function Φ 0 (M 0 ) describes the radial part of the configuration. Note that like the TISM configurations s 3 , s 2 p, . . . etc. the respective LF configurations have a common radial part which is the same as the radial wave function Φ 0 (M 0 ) of the ground state configuration s 3 . The normalization factor N where k is a relative momentum defined in Eqs. (3) and (A10) -(A12). Dots in the curly brackets denote the same expression as in the curly brackets of Eq. (11). The LF state vector analogous to the TISM configuration s 2 p[21] X (l = 1)y (2) X is defined by an equation similar to Eq. (12): In the case of l = 0(l 12 = l 3 = l) the basis vectors |s 3 [3] X [M 0 , j(l = 0, s(s 12 ))];P , µ j f and the state vector |[3] X [M, j(l = 0, s(s 12 ))];P , µ j f are also defined by Eqs. (11) and (12), respectively, but with the other value of l = 0 and with the spherical wave function Y 00 = 1 4π . In this case the radial part of the LF configuration s 3 [3] X (a nucleon, the ground state) is described by the wave function Φ 0 (M 0 ). The radial part of the excited LF configuration , where a free parameter c 2 is chosen to satisfy the orthogonality condition N * 1/2 + |N = 0. The main drawback of the configurations |s 2 p[21] X y (1) X and |s 2 p[21] X y (2) X defined as orbital states l = 1(l 12 = 0, l 3 = 1) and l = 1(l 12 = 1, l 3 = 0) is that the partial waves l 3 = 1 and l 12 = 1 of the basis vectors are defined (Eqs, (13) and (11)) in different reference frames. The angular momentum l 3 = 1 is defined in the CM frame, while the state with angular momentum l 12 = 1 is defined in the rest frame of the two-quark cluster. Such a difference presents difficulties in constructing state vectors satisfying the Pauli exclusion principle. In the final step of the construction of a fully symmetric state [3] XST one should reduce the product of two irreducible representations of the S 3 group, [21] X and [21] ST . Both orbital states, [21] X , should be defined in a common reference frame, e.g. in the CM, otherwise it will be impossible to use a standard technique of reducing the product of two irreducible representations.
To solve the problem we start from basis vector |[21] X y (1) X defined in the CM by Eq. (11). We construct the second basis vector |[21] X y (2) X of this irreducible representation given in the CM using pairwise permutations P ij of quarks in the r.h.s. of Eq. (11). Doing so we have obtained a new linear-independent component of the basis of the given irreducible representation, which we denote as |[21] X y (2) X CM . The new basis vector is represented by a modified Eq. (13) in which the angular part of the integrand has been transformed into the function (κ/β) l Y lµ l (κ). It depends on a modified momentum κ, where k z = 1 2 (k + − k − ). Starting from the relations . . , etc., one can verify that the matrix elements of quark permutations P ij between new basis states |[21] X , y (1,2) X CM are equal to the standard values characteristic of the given irreducible representation of the group S 3 [58].

IV. CURRENT MATRIX ELEMENTS IN QUARK REPRESENTATION
A. Spin-orbital part of the matrix element At light front, the plus-component of the current I + (x) = J 0 (x) + J 3 (x) alone is sufficient to determine the full set of observables including the transition form factors (if the current satisfies the continuity equation ∂ ν J ν = 0). In addition, the current matrix element for a Dirac particle between front states (A3) does not depend on particle momenta at all, Only if the quark has an anomalous magnetic moment κ i , a term depending on the momentum transfer q ν = p ′ i ν − p ν i arises. In the Breit frame, where q ν = {0, q ⊥ , 0, 0}, a general one-particle current matrix element is of the form where e i = 1 6 + 1 2 τ iz is the quark charge.
In the case of reaction N + γ * → N * the transition matrix element of the quark current (16) between nucleon and baryon state vectors can be readily represented in a special Breit (B) frame, where the momenta of the initial nucleon (P B ) and the final baryon (P ′ B ) are equal with and The desired matrix elements f [21] X , y (n) where the initial nucleon is represented by the ground state configuration |[3] X [M, j(s(s 12 ))];P , µ j f and the final baryon is described by the configurations defined in Eqs. (12) (n = 1) and (14) -(15) (n = 2), have been reduced to six-dimensional integrals over invariant light-front variables K ⊥ , k ⊥ , ξ and η, The momentum k ′(n) takes the value depending on the index n = 1 or 2 (i.e. the value of Ymanouchi symbol y (n) X ), as it follows from (11) and (15).
is a Jacobian. We use the notation I s12 ({k ′ i },{k i }) for the one-particle current matrix element (16) of the third quark, which is modified by Clebsch-Gordan coefficients used for adding spins and by the D matrices of the Melosh transformations, as follows from Eqs. (11) -(14), Eqs. (18) and (19) are only written for the current of the third quark, but we use combinatoric factor 3 that allows to take into account contributions of all 3 quarks. In Eqs. (18) and (19) the primed symbols indicate that they are of the final state wave functions. It is important that only the momentum K ⊥ from all the full set of independent momenta in the three-quark system (K ⊥ , k ⊥ , ξ, η) changes its value for the absorption of a photon, K ′ ⊥ = K ⊥ + ηq ⊥ . As a result, the individual momenta of quarks in the CM frame take the values k ′ The direct calculation of spin sums in Eq. (19) results in a rather cumbersome expression depending on the zcomponents of the total 3q spin s = where the coefficients dA, dB, dC and dD depend on q ⊥ and on the inner momenta K ⊥ , k ⊥ , ξ, η. The expansion in Eq. (20) is a generalization of the analogous expansion for the one-particle quark current (16), where . The full series in the r.h.s. of Eq. (20) includes all the spin structures which contribute both to transitions without parity change (I and iσ 2 ) and with a change in parity (σ 1 and σ 3 ). The integration of the effective current (20) in a product with the spherical functions in the r.h.s. of Eq, (18) and the convolution with "spin-orbital" Clebsch-Gordon coefficients over indices µ ′ l , µ ′ s leads to two different types of transition matrix elements: 1) for transitions with parity change (l = 1) and a corresponding representation 2) for transitions without a change in parity (l = 0) with the only difference that at l = 0 there is no dependence on the Yamanouchi symbol y (n) and The functions A s12 , B s12 , C s12 represent the full set of necessary spin-orbital matrix elements to compose a final expression for the transition/elastic amplitude, but to do so isospin must be taken into account. The final expression should be a linear combination of these functions with coefficients depending on the isospin matrix elements (see below).

B. Radial part
It should be realized that the expression for the spin-orbital matrix element given in Eq, (19) is also true in the case of a positive parity final state. Then the angular part of final wave function k ′(n) β l Y * lµ ′ l (k ′(n) ) has to be changed to a constant Y 00 = 1 4π , the Yamanuchi symbols should be omitted and we substitute the function for N 1 Φ 0 (M ′ 0 ). Function (24) is the analogy of the TISM wave function |sp 2 [3] 2β ξ ] (k and K are non-relativistic momenta, β ξ and β ρ are the respective scales). Similarly, in the case of the elastic process N + γ * → N the radial part of the ground state wave function N 0 Φ 0 (M ′ 0 ) should be substituted into Eq. (19) instead of the resonance radial part. Here we use the pole-like function [42] which gives a good description of elastic nucleon form factors [18,42] in a wide interval of Q 2 , where data exist, 0≤ Q 2 32 GeV 2 . The relative values of u-and d-quark contributions to the form factors are also well described in this model [59]. One might expect (and this is supported by our calculations, see below) that the transition form factors of the low-lying nucleon resonances can be described by a common function (25) for both the nucleon and the resonances. However, the use of function (25) in the case of the Roper resonance leads to an overestimate of the transition form factors [18], at least in the region of moderate/high values of Q 2 1 -2 GeV 2 . This possibly means that the Roper resonance is a more loosely bound system than the nucleon. Such an assumption is well correlated with the results [17,18,33,53] obtained with modified wave functions for the resonance. The question arises as to whether there is a soft 3q component of the Roper resonance. Otherwise the standard (hard) 3q wave function should be modified by the addition of a soft hadronic component [18]. In an effort to test these hypotheses we consider here a "hybrid variant" of the wave function Φ 0 for the Roper resonance, with a considerable weight for the Gaussian component (α ≃ 0.25 − 0.5).The Gaussian adds a loose 3q component to the pole-like wave function (25).

C. Isospin and the Pauli exclusion principle
From the Pauli exclusion principle which requires the use of fully antisymmetric state vectors in initial and final states, it would be convenient to rewrite all the transition matrix elements in terms of Young schemes and Yamanouchi symbols. Initial and final states in the matrix elements (22) and (23) are given, in fact, in the required form, since the value of the total spin s = 1 2 corresponds to the Young scheme [21] S , while the value of the spin of a two-particle subsystem, s 12 = 1 and 0, corresponds to the Yamanouchi symbols y Here the Young schemes [21] S and [21] T are omitted to minimize the complexity of notations. The value of this matrix element is a product of the charge matrix element S , S .
and use a trivial relation | [21] XS , y (n) for the initial state. In the final step we take into account the isospin T and define a fully symmetric state with the Young scheme [3] XST in the united XST space, T , D. Transition amplitudes with/without change of parity We started from the general expressions (31)-(32) for the transition amplitudes of the reactions N 1/2 + + γ * → N * J P , which was obtained in the framework of a LF quark model in the special Breit frame (17). We also derived the following representations (in terms of the functions A, B, C and D defined in Eqs. (22)-(23)) for the (elastic and inelastic) amplitudes and transition form factors: 1) elastic scattering where (the function (25) is used in the calculation of A s12 (q ⊥ ) and B s12 (q ⊥ )); 2) transition without parity change N 1/2 + + γ * → N * where (the modified function (26) is used in the calculation of A s12 (q ⊥ ) and B s12 (q ⊥ )); 3) transition with a change of parity (the function (25) is used in the calculation of D s12 (q ⊥ ) and C s12 (q ⊥ ))

A. Form factors
The form factors f i , f R i andf i are invariant functions which can be used to describe observables in any reference frame. The calculated observables (cross sections, helicity amplitudes, . . . etc.) should therefore be independent on the forms of the dynamics. Hence, one can describe an observable, e.g. the helicity amplitude, by using the nucleon current in the instant form, and transform the current matrix elements to the light front without changing the value of the observable. This can be used to relate the LF quark form factors f i , f R i ,f i to standard ones F i , F R i ,F i used in the parametrization of the instant nucleon current.
Here we consider the plus-component of the nucleon current (39), J + N = J 0 N +J 3 N , as a matrix element of an operator I + N (0) between initial and final states represented by Dirac spinors u N (p) andū N * (p ′ ) (note the quark operator I + (0) has been defined by just the same method). It follows from Eq. (39) that the operators which generate transitions with parity change (Ĩ + N ) or without (I + N ) are of the form Starting from the matrix elementsū N * (p ′ )I + N (0)u N (p) andū N * (p ′ )Ĩ + N (0)u N (p) written in the special Breit frame (17) we transform the initial/final states to state vectors at light front using the Melosh transformation (8) - (9). In the end we obtain the LF matrix elements of the nucleon current parametrized by the form factors F i , F R i ,F i : 1) elastic scattering 2) transition without parity change N 1/2 + + γ * → N * 3) transition with a change in parity Comparing the matrix elements of the nucleon current of Eqs. (41) - (43) to the LF quark model predictions given by Eqs. (33) - (38) one can see that both parametrizations of the transition/elastic form factors, Thus the form factors f i , f R i ,f i , which are related to the functions A, B, C and D by Eqs. (34), (36) and (38) respectively, give definite predictions for the observables F i , F R i ,F i .

B. Helicity amplitudes
We use the standard definitions (PDG [60]) for the transverse (A 1/2 ) and longitudinal (S 1/2 ) helicity amplitudes, written in the resonance CM frame (CM momenta are denoted by an asterisk, p * = −q * ), In above equations K w = Substituting the nucleon current (39), parametrised by the form factors F R i andF i , into the r.h.s. of Eq. (45) one obtains [17,20,61,62] expressions for the desired helicity amplitudes: 1) for the electroproduction of positive parity resonances, 2) for the electroproduction of negative parity resonances,

VI. RESULTS AND CONCLUSIONS.
We study the electroproduction of low-lying nucleon resonances in the framework of a relativistic quark model. Quark configurations at light front are developed here for orbitally/radially excited states satisfying the Pauli exclusion principle. The next step in the study could be, in analogy to the nuclear shell model, to take into account configuration mixing. In hadron physics, however, it would be more effective to take into account a non-quark component of the baryon considering the lowest quark configurations as the "quark core" of the resonance while adding higher Fock states, e.g. a "meson cloud".
Previously we have used another important ingredient of our approach -the hadron molecule model [18,51,63], which allows to represent effectively a hadronic component of the resonance. The corresponding technique has been firstly sugested and thereafter developed in Refs. [64] for the description of certain hadronic resonances dropping out from the standard quark model classification.
In the first approximation a baryon resonance can be represented as a mixed state of the quark core (3q * ) and the hadron molecule (B + M ), The hadron molecule as a loosely bound B + M state can only give a "soft" contribution to the transition amplitude. This contribution should be important at small/moderate values of Q 2 1 − 2 GeV 2 , e.g., in the case of Roper resonance [51], where the helicity amplitude A 1/2 crosses the zero value at Q 2 ≃ 0.5 GeV 2 . In the region of high momentum transfers the contribution of the hadron molecule to the transition form factors approaches zero, and can be neglected. It should be mention that this component has a weight of sin 2 θ in the normalization integral, and thus the observable contribution of the quark core to the form factors will be reduced as compared with the ordinary quark model prediction. This should be taken into account when one compares the quark model results to data at high Q 2 . A possible underestimate of the quark model predictions to the data can lead to an estimate for the mixing angle θ.
Our results for the transition form factors and helicity amplitudes are shown in Figs. 1 and 2 in comparison with the high-quality data of the CLAS collaboration [1][2][3][4][5][6][7][8][9][10]12]. We have used only three free parameters, m, β and γ, in the wave function of quark core Φ 0 of Eq. (25) in both cases the nucleon and the negative parity resonance. Here m = m 1 = m 2 = m 3 is the mass of a light constituent quark (u, d). In the case of the Roper resonance we also use two additional parameters, the coefficients α and β 1 of Eq. (26). The parameters m, β and γ are common to all the resonances. The parameter c 2 in the wave function of the radially excited quark core (24) is not free, since it is determined from the orthogonality condition Φ 0 |Φ 02 = 0. We neglect the quark anomalous magnetic moments (κ 1 = κ 2 = κ 3 = 0) in the quark current defined in Eq. (16) as their values are too small (κ i 0.03, according to Ref. [18]). The only influence they have is on the precise value of the baryon magnetic momentum. Parameters m, β and γ are taken from Refs. [18,42] where they were fitted to the nucleon data in a large interval of 0≤ Q 2 32 GeV 2 . Only the coefficients α and β 1 for a superposition of the Gaussian and the pole-like wave function of Eq. (26) has been varied to obtain the best description of the Roper resonance helicity amplitudes. In the end we obtain a decent description (Figs. 1 and 2) of form factors and helicity amplitudes of the three baryons (including the elastic nucleon form factors described in Ref. [18]) at moderate/high momentum transfer, Q 2 1-2 GeV 2 , making use of the following values of parameters: In the latter case one can improve the agreement by using a large mixing angle θ R for the molecular state N + σ in Eq. (48), taking e.g. cos θ R ≃ sin θ R ≈ 0.7 as we have done in our previous work [18]. However, the most realistic variant is a large mixing parameter for another (loose) quark configuration given in Eq. (26), (thin small-dashed lines in Fig. 2 which correspond to the value of α = 0.245). Then we obtain a good agreement with the data for the both resonances, N 1/2 − (1535) and N 1/2 + (1440), using small values of the mixing angle for the respective molecular states, cosθ ≃ cos θ R ≃ 0.93-0.95 ( Figs. 1 and 2, solid lines). The shaded region in Fig. 2 (left upper panel) shows the range of the Roper helicity amplitude A 1/2 with the mixing angle changing from θ R = 0 to θ R = 18 0 . Our results demonstrate that the contribution of the hadron molecule to the transition amplitude quickly dies out with rising Q 2 and might be neglected at high Q 2 . The contribution of the quark core correlates well with the data at Q 2 1-2 GeV 2 , if the parameter of mixing cos θ is about 0.93-0.95. On this basis we predict the Q 2 -behavior of amplitudes at high Q 2 5-7 GeV 2 starting from the quark core wave function alone.
In the case of the Roper resonance there are discrepancies between the predictions of the model with the pole-like wave function Φ 0 (dashed curves in Fig. 2) and the data. We have shown that one can considerably improve the agreement with data modifying only the quark core wave function by the replacement Φ 0 → Φ R 9 following Eq. (26). This can be considered as an argument in support of the inner quark structure of the Roper resonance contrary to what might be expected from the above mentioned large discrepancies between predictions and data.
It is possible that in the case of the Roper resonance the unknown multiparticle component of the quark current plays a more important role than in the case of other resonances. It can be instructive to compare the results of our model (solid curves in Figs. 1 and 2) with a good description of the Roper resonance transition form factors recently obtained in Ref. [33] (dotted curves in Fig. 2) in a soft-wall AdS/QCD. The results of both LF models are close to each other (and close to the data) at Q 2 1-2 GeV 2 , but at low Q 2 the results of the LF quark model considerably differs from the AdS/QCD results which stay close to the CLAS data. This discrepancy can especially be traced to the strict requirement of orthogonality for the ground (0S) and excited (2S) radial wave functions of the N and R states belonging to quark configurations with the same spin-isospin (S = 1/2, T = 1/2) and symmetry ([3] ST [3] X ) quantum numbers. Then, for the transition N → R (Roper), the matrix element of the single-particle current (16), which does not act on the orbital part of the wave function, should vanish for Q 2 → 0 (because of the orthogonality of the orbital parts of the baryon wave functions R orb |N orb = 0), as it really seen in Fig. 2 (solid and dashed curves are close to zero at Q 2 → 0). But the A 1/2 data at Q 2 ≃ 0 are not small. Instead they cross the Q 2 axis at Q 2 ≃ 0.5 GeV 2 . The discrepancy of the quark model results and the data in this region can be an effect of multiparticle currents. We have modeled such an effect in our preceding work [51] using a non-relativistic 3 P 0 model for vacuumqq pairs. As a result we have obtained a realistic description of the amplitude A 1/2 at small values of Q 2 (dotted-dashed curves in Fig. 2). In the region of Q 2 1-2 GeV 2 , where a non-relativistic quark model can be used reliably, such descriptions are very close to the CLAS data. In both cases of the N → N 1/2 + (1440) and N → N 1/2 − (1535) transitions AdS/QCD approach [33,35] (dotted lines) gives very good discription of data and at large Q 2 it is very close to the the LF quark model results (solid lines). Note that successfull descrption of data in AdS/QCD approach in low energy domain is explained by inclusion of higher Fock states contribution into the structure of nucleon and nucleon resonances, while at high Q 2 it is provided by the correct power scaling of the form factors/helicity amplitudes consistent with quark counting rules. Summarizing the results shown in Figs. 1 and 2 it is worth noting that a good basis of quark configurations constructed at the light front, as performed in Sect. III-IV, might be an effective tool in the study of the inner structure of baryons. This is particularly true when the study is based on high-quality data on the baryon electroproduction at high momentum transfer.
Apart from the canonical boost, the momenta p ′µ i and p µ i can be connected by another element G of the homogeneous Lorentz group. In particular, it might be the "front boost" Λ f (P M0 ) with the respective front basis |(m i , s i ); which are used in the front form of the dynamics. The matrices Λ f which connect the momentap ′ i andp i ,p ′μ i = Λ f (P M )μ νpν i , are elements of the "front subgroup" of the homogeneous Lorentz group. The light front t − z = 0 is invariant under transformations of the front subgroup.
Canonical boosts Λ( P M0 ) itself do not form a subgroup of the Poincaré group, since the product of two boosts Λ c gives rise to the Wigner rotation R w while the product of front boosts does not give rise to the Wigner rotation, Starting from Eqs. (A1) and (A4) and using the relationship RΛ c ( pi mi )R −1 = Λ c ( Rpi mi ) [49,50] one readily obtains that the unitary irreducible representation of the canonical boost in the free basis (A1) is of the form where the arguments of the D function are the Euler angles of the Wigner rotation (A4). The unitary irreducible representation of the front boost is of a trivial form According to Eq. (A7) the z component of the front spin is a kinematical variable with the value of µ i being constant at any transformation which leaves the light front t − z = 0 invariant (including the spatial rotations around the z axis). Therefore the µ i can be identified with an additive quantum number, the helicity of the particle at the light front [57]. In Eqs. (A6) and (A7) the connection between the momenta k i and p i is symbolically written as p i = Λ k i . This implies the following 4 × 4 matrices for boosts Λ c and Λ f [49,50]: The front boost (A9) does not mix the 'kinematical component" , while the canonical boost (A8) mixes the k i and the ω i (k i ). However in both cases the 3-momentum is an additive quantum number: P + 12 = p + 1 + p + 2 , P + = p + 1 + p + 2 + p + 3 , P 12⊥ = p 1⊥ + p 2⊥ , . . . , etc. A similar property holds for the relative momenta,k = Λ −1 f (P 12 m12 )p 1 andK ≡k 3 = Λ −1 f (P M0 )p 3 , which are connected with the momentap i by the linear relations k + = m 12 P + 12 p + 1 = where These relations can be readily obtained by using the inverse of the matrices (A9). Since x 1 + x 2 + x 3 = 1, only two independent parameters, ξ and η, instead of x 1 = ξη, x 2 = (1−ξ)η and x 3 = 1−η are used. The important property of the variables (A10) -(A12) is that the values of k ⊥ = |k ⊥ |, K ⊥ = |K ⊥ | and x i are relativistic invariants (it can be readily verified with the relations (A9)) and the invariant masses m 12 and M 0 defined in Eq. (4) are functions only of k ⊥ , K ⊥ , ξ and η, In particular, one can use the function M 0 (k ⊥ , K ⊥ , ξ, η) as an argument of the relativistic wave function Φ LF Mj (M 0 ) in Eq. (5) rewritten at the light front.  [2], squares (bold) [5], triangle (bold) [6], circles (empty) [7], triangles (empty) [8], squares (empty) [9], diamonds [10]. Theoretical description in terms of a light front (LF) quark model: dashed curves -results of calculations on the basis of three-quark configurations, s 2 p for the N * and s 3 for N , using a pole-like quark core wave function Φ0(denoted by LFQM(P) in the legends); solid curves -results for the model of Eq. (48) with the "strong" value of mixing parameter cos θ = 0.951 (θ = 0.1π); dotted curves -results of the soft-wall AdS/QCD model [35]. for N (940), using a pole-like quark core wave function Φ0; short dashed curves -results of calculations using a superposition given in Eq.(26) of pole-like function (P) and Gaussian (G) in the quark core wave function Φ R 0 with α =0.245 (denoted by LFQM(P+G) in the legends); solid curves -results for the mixed-state model given in Eq. (48) with a "strong" value of the mixing angle, cos θR = 0.930 (θR = −0.12π) (denoted by LFQM(P+G)+N σ in the legends); dotted curves -results obtained in the soft-wall AdS/QCD approach [33]; dashed-dotted curves -results for the nonrelativistic quark model (NRQM) with contributions of virtualqq pairs in terms of the 3 P0 model [51]. The CLAS data (bold and empty circles) on one-pion [2,17] and two-pion [3] electroproduction off the proton. The A1 data on π 0 (the empty square) electroproduction [11].